The accurate numerical prediction of large-amplitude nonlinear ship motions is quite important for the safe navigation of ships under rough sea conditions. A weakly nonlinear ship motion model is proposed to analyze the vertical ship motion in the wave. In the numerical model, the radiation force and the diffraction force are estimated by using the impulse response function, and the ship slamming force is introduced according to the momentum impact theory. The incident wave forces (Froude-Krylov forces (F-K)) and hydrostatic forces on the instantaneous wetted ship surface are calculated for ship under the incident wave profile. One of the unique features is that the fully nonlinear stream function wave theory is introduced to estimate the incident wave kinematics, dynamic pressure and the F-K fluid loads, where the detailed numerical results are presented and discussed. The proposed numerical model is used to estimate the vertical nonlinear ship motions of a S175 container ship sailing in regular waves. It is found that the vertical FK forces and the heave motions based on the stream function wave theory are slightly larger than those of the linear wave theory especially when the incident wave steepness is relatively large. Another feature is that the momentum impact theory is used to estimate the section slamming forces. The results show that at a relatively high speed, the slamming force greatly impact the ship vertical motion, and the inclusion of the slamming force contributes evidently to the reduction of the motion amplitude.